Optimal vaccinations: Cordons sanitaires, reducible population and optimal rays

TL;DR

This paper explores optimal vaccination strategies in complex populations, balancing cost and infection control, and introduces graph-theoretic and spectral methods to identify effective vaccination policies.

Contribution

It provides explicit formulas and conditions for optimal vaccination strategies in infinite-dimensional populations, challenging the effectiveness of cordon sanitaire approaches.

Findings

Cordon sanitaire may not be optimal but outperforms worst strategies.

Explicit formulas for best and worst vaccination strategies are derived.

Conditions for scaling optimal strategies to remain optimal are established.

Abstract

We consider the bi-objective problem of allocating doses of a (perfect) vaccine to an infinite-dimensional metapopulation in order to minimize simultaneously the vaccination cost and the effective reproduction number $R_e$, which is defined as the spectral radius of the effective next-generation operator. In this general framework, we prove that a cordon sanitaire, that is, a strategy that effectively disconnects the non-vaccinated population, might not be optimal, but it is still better than the "worst" vaccination strategies. Inspired by graph theory, we also compute the minimal cost which ensures that no infection occurs using independent sets. Using Frobenius decomposition of the whole population into irreducible sub-populations, we give some explicit formulae for optimal ("best" and "worst") vaccinations strategies. Eventually, we provide some sufficient conditions for a scaling of an optimal strategy to still be optimal.